Hamel bases and well–ordering the continuum

In ZF, the existence of a Hamel basis does not yield a well–ordering of R. Throughout this paper, by a Hamel basis we always mean a basis for R, construed as a vector space over Q. We denote by E the Vitali equivalence relation, xEy iff x− y ∈ Q for x, y ∈ R. We also write [x]E = {y : yEx} for the E–equivalence class of x. A transversal for the set of all E–equivalence classes picks exactly one member from each [x]E . The range of any such transversal is also called a Vitali set. A set Λ ⊂ R is a Luzin set iff Λ is uncountable but Λ ∩M is at most countable for every meager set M ⊂ R. A set S ⊂ R is a Sierpiński set iff S is uncountable but S ∩ N is at most countable for every null set N ⊂ R (“null” in the sense of Lebesgue measure). A set B ⊂ R is a Bernstein set iff B ∩P 6= ∅ 6 = P \B for every perfect set P ⊂ R. It has been well–known for more than a century that the existence of a well– ordering of the reals implies the existence of all these “pathological” sets of reals: 1The first author gratefully acknowledges support from the Marianne und Dr. Horst KiesowStiftung, Frankfurt a.M. 2The second author was partially supported by the SFB 878 “Groups, geometry, and actions” from the DFG (Deutsche Forschungsgemeinschaft). Most of the results in this paper were obtained while the second author was an invited visiting researcher at the Centre de Recerca Matemàtica (CRM) in Bellaterra (Catalonia), and he would like to thank the CRM for their support and hospitality. This paper was typed while the second author visited Grigor Sargsyan in Warsaw (Poland) in Nov 2016, whom he would like to thank for his hospitality and support. He would also like to thank William Chan for his comments on an earlier version of this paper. 3The third author would like to acknowledge the support through the funding Projects NSFC 11321101 and 11401567. 4The fourth author gratefully acknowledges support from the National Natural Science Fund of China, No. 11322112 and 11671196 and from a Humboldt Research Fellowship for Experienced Researchers.